Hurwitz spaces, Nichols algebras, and Igusa zeta functions

Abstract: By constructing new quasimap compactifications of Hurwitz spaces of degrees 4 and 5, we establish a new connection between arithmetic statistics, quantum algebra, and geometry and answer a question of Ellenberg-Tran-Westerland and Kapranov-Schechtman. It follows from the geometry of our compactifications and a comparison theorem of Kapranov-Schechtman that we can precisely relate the following 3 quantities: (1) counts of $\mathbb{F}_q[t]$-algebras of degrees 3, 4, and 5 (2) the ``invariant'' part of the cohomology of certain special Nichols algebras (3) Igusa local zeta functions for certain prehomogeneous vector spaces. Using Igusa's computation of the zeta function for the space of pairs of ternary quadratic forms, we compute the number of quartic $\mathbb{F}_q[t]$-algebras with cubic resolvent of discriminant $q^b$ and the part of the cohomology of a 576-dimensional Nichols algebra $\mathfrak{B}_4$ invariant under a natural $\mathbf{S}_4$-action. From the comparison for degree 3, we also obtain two answers to Venkatesh's question about the topological origin of the secondary term in the count of cubic fields.